Problem: Find an explicit formula for the geometric sequence $\dfrac12\,,-4\,,\,32\,,-256,..$. Note: the first term should be $\textit{a(1)}$. $a(n)=$
Solution: In a geometric sequence, the ratio between successive terms is constant. This means that we can move from any term to the next one by multiplying by a constant value. Let's calculate this ratio over the first few terms: $\dfrac{-256}{32}=\dfrac{32}{-4}=\dfrac{-4}{\frac12}={-8}$ We see that the constant ratio between successive terms is ${-8}$. In other words, we can find any term by starting with the first term and multiplying by ${-8}$ repeatedly until we get to the desired term. Let's look at the first few terms expressed as products: $n$ $1$ $2$ $3$ $4$ $h(n)\!\!\!\!\!$ ${\dfrac12}\cdot\!\!\!\left({-8}\right)^{\,0}\!\!\!\!\!\!$ ${\dfrac12}\cdot\!\!\!\left({-8}\right)^{\,1}\!\!\!\!\!\!$ ${\dfrac12}\cdot\!\!\!\left({-8}\right)^{\,2}\!\!\!\!\!\!$ ${\dfrac12}\cdot\!\!\!\left({-8}\right)^{\,3}$ We can see that every term is the product of the first term, ${\dfrac12}$, and a power of the constant ratio, ${-8}$. Note that this power is always one less than the term number $n$. This is because the first term is the product of itself and plainly $1$, which is like taking the constant ratio to the zeroth power. Thus, we arrive at the following explicit formula (Note that ${\dfrac12}$ is the first term and ${-8}$ is the constant ratio): $a(n)={\dfrac12}\cdot\left({-8}\right)^{{\,n-1}}$ Note that this solution strategy results in this formula; however, an equally correct solution can be written in other equivalent forms as well.